The PCAN method nonparametrically characterizes network samples via subgraph count statistics. Subgraph counts and related network summaries are commonly used in descriptive analysis to explore the properties of network-valued data; such analyses are well studied (see for example [1]}, [2]}, [3]}). Subgraph counts are also commonly treated as features in generative models for network-valued data. The family of exponential random graph models, for example, use subgraph counts as sufficient statistics for an observed network [4]}. Similar analyses in [5]} established the sufficiency of subgraph counts under several popular classes of generative models in the family of kernel-based random graphs, including the stochastic block model [6]}, [7]}, the Erdős-Rényi random graph model [8]}, and the latent space model [9]}. The large-sample analysis of the PCAN method builds on the foundations established in [5]} as well the first order analyses of subgraph counts in [11]}. In our current work, we analyze use of subgraph counts as sufficient statistics under the kernel-based random graph model when the objective is to identify principal components of the network sample.
